# Computer Simulation of Temperature Distribution during Cooling of the Thermally Insulated Room

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## Abstract

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## 1. Introduction

## 2. Model Description

- Specified temperature (${T}_{s}$) on a surface: ${T}_{s}=T\left(x,y,z,t\right).$
- Heat flow density $\left({q}_{s}\right)$ on a surface: ${q}_{s}=q(x,y,z,t).$
- Convective boundary condition: ${q}_{x}{n}_{x}+{q}_{y}{n}_{y}+{q}_{z}{n}_{z}=h\left({T}_{s}-{T}_{e}\right)+{q}_{r}.$

#### 2.1. Simulation Conditions

- Room dimensions: length 8.7 m, width 5 m, height 3 m.
- The initial temperature of the air inside the tested room $21{\phantom{\rule{0.166667em}{0ex}}}^{\xb0}\mathrm{C}$.
- Temperature of air in the adjoining rooms of the building $0{\phantom{\rule{0.166667em}{0ex}}}^{\xb0}\mathrm{C}$.
- Velocity of the airflow inside the tested room $0.03\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\xb7{\mathrm{s}}^{-1}$.
- Heat transfer coefficient inside the building $8\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{K}}^{-1}$.
- Heat transfer coefficient outside the building $25\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{K}}^{-1}$.

#### 2.1.1. Domain Setting

#### 2.1.2. Boundary Setting

#### 2.2. Computer Model Construction

- Defining the dimensions and location of all model elements.
- Defining the physical properties (specific heat capacity, thermal conductivity, density) of all model elements.
- Defining the initial and boundary conditions.
- Solving the model under modified conditions.
- Displaying the results as both 3D and 2D plots and export the output data into the TXT files.

#### 2.3. Simulation Setup

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${C}_{p}$ | Heat capacity, $\left[\mathrm{J}\xb7{\mathrm{K}}^{-1}\right]$; |

${c}_{p}$ | Specific heat capacity, $\left[\mathrm{J}\xb7{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\right]$; |

h | Heat transfer coefficient, $\left[\mathrm{W}\xb7{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}\right]$; |

n | Normal vector, $\left[-\right]$; |

q | Heat flow density, $\left[\mathrm{W}\xb7{\mathrm{m}}^{-2}\right]$; |

${q}_{r}$ | Incident radiant heat flow per unit surface area, $\left[\mathrm{W}\xb7{\mathrm{m}}^{-2}\right]$; |

${q}_{s}$ | Heat flow density on the surface, $\left[\mathrm{W}\xb7{\mathrm{m}}^{-2}\right]$; |

t | Time, $\left[\mathrm{s}\right]$; |

${t}_{D}$ | Time to achieve desired temperature decrease, $\left[\mathrm{h}\right]$; |

T | Temperature, $\left[\mathrm{K}\right]$; |

${T}_{amb}$ | Ambient temperature, $\left[\mathrm{K}\right]$; |

${T}_{inf}$ | External temperature, $\left[\mathrm{K}\right]$; |

${T}_{e}$ | Convective exchange temperature, $\left[\mathrm{K}\right]$; |

${T}_{s}$ | Surface temperature, $\left[\mathrm{K}\right]$; |

${T}_{0}$ | Initial temperature of a body, $\left[\mathrm{K}\right]$; |

v | Fluid velocity, $\left[\mathrm{m}\xb7{\mathrm{s}}^{-1}\right]$; |

x, y, z | Space coordinates, $\left[\mathrm{m}\right]$; |

$\lambda $ | Thermal conductivity, $\left[\mathrm{W}\xb7{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right]$; |

$\mathsf{\Phi}$ | Inner heat-generation rate per unit volume, $\left[\mathrm{W}\xb7{\mathrm{m}}^{-3}\right]$; |

$\epsilon $ | Emissivity, $\left[1\right]$ |

$\sigma $ | Stephan–Boltzmann constant, $\sigma =5.670367\xb7{10}^{-8}\phantom{\rule{0.277778em}{0ex}}\mathrm{W}\xb7{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ |

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**Figure 2.**Temperatures of air in the adjoining rooms and the temperature of the air outside the building—data used as the boundary conditions for the tested model.

**Figure 3.**Meshed model: (

**a**) sketch of the model; (

**b**) close-up view of a section near the outer wall; and (

**c**) percentage temperature deviations related to the mesh densities.

**Figure 4.**Time evolution of the air temperature in the center of the room: (

**a**) sketch of the monitored location (the red point) inside the room; (

**b**) dependence of the air temperature on time of cooling the room.

**Figure 5.**Temperature and heat flow in the outside wall after two days of cooling the room: (

**a**) position of the monitored place (the red line); (

**b**) temperature and heat flow distribution.

**Figure 6.**Slices of temperature inside the cooled room: (

**a**) initial temperature distribution; (

**b**) temperature distribution after one day of cooling.

**Figure 7.**Distribution of the air temperature in selected times of cooling the room: (

**a**) the spatial location of the plane in which the temperature was determined; (

**b**) slices with the temperature distributions.

**Figure 8.**Time evolution of the air temperature in the middle of the room concerning the specific heat capacity of the outside wall thermal insulation as (

**a**) 3D plot and (

**b**) 2D temperature distribution.

**Figure 9.**Deviation of the air temperature while cooling the thermally insulated room from the room with non-insulated outside wall as (

**a**) 3D plot and (

**b**) 2D temperature distribution.

**Figure 10.**Determination of desired temperature decrease from time evolution of the air temperature inside the cooled room based on approximation of output data from COMSOL Multiphysics by polynomial of the 6th degree.

Geometrical Element | Size $\left[\mathbf{m}\right]$ | Thermal Conductivity $\left[\mathbf{W}\xb7{\mathbf{m}}^{-1}\xb7{\mathbf{K}}^{-1}\right]$ | Density $\left[\mathbf{kg}\xb7{\mathbf{m}}^{-3}\right]$ | Specific Heat Capacity $\left[\mathbf{J}\xb7{\mathbf{kg}}^{-1}\xb7{\mathbf{K}}^{-1}\right]$ | Emissivity $\left[1\right]$ |
---|---|---|---|---|---|

Rear wall | $8.7\times 0.3\times 3$ | 0.27 | 900 | 960 | 0.85 |

Left side wall | $5\times 0.3\times 3$ | 0.27 | 900 | 960 | 0.85 |

Right side wall | $5\times 0.3\times 3$ | 0.27 | 900 | 960 | o.85 |

Wall under the windows | $8.4\times 0.3\times 1.05$ | 0.80 | 1700 | 900 | 0.80 |

Wall above the windows | $8.4\times 0.3\times 0.75$ | 0.80 | 1700 | 900 | 0.80 |

Columns in the outside wall | $0.3\times 0.3\times 3$ | 0.32 | 2192 | 1018 | 0.85 |

Floor | $8.7\times 5\times 0.3$ | 1.43 | 2300 | 1020 | 0.85 |

Ceiling | $8.7\times 5\times 0.3$ | 0.82 | 1251 | 1021 | 0.8 |

Insulation above the ceiling | $8.7\times 5\times 0.3$ | 0.04 | 30 | 1270 | 0.90 |

Insulation of the outside wall | $8.7\times variable\times 1.05$ | 0.05 | 3000 | 1500 | 0.85 |

Window frames | $4.2\times 0.145\times 1.2$ | 0.18 | 400 | 2510 | 0.89 |

Glass in larger windows | $1.2\times 0.008\times 0.8$ | 0.76 | 2600 | 840 | 0.99 |

Glass in smaller windows | $0.6\times 0.008\times 0.8$ | 0.76 | 2600 | 840 | 0.99 |

Door leaf | $1\times 0.05\times 2.2$ | 0.11 | 800 | 1150 | 0.89 |

Panel next to the door | $0.65\times 0.05\times 2.2$ | 0.20 | 1380 | 1100 | 0.94 |

Door frame | $1.85\times 0.1\times 2.3$ | 58 | 7850 | 440 | 0.89 |

Door trim | $0.45\times 0.05\times 1.8$ | 0.76 | 2600 | 840 | 0.9 |

Left heater | $0.2\times 0.5\times 1$ | 58 | 7850 | 440 | 0.88 |

Right heater | $0.2\times 0.5\times 1$ | 58 | 7850 | 440 | 0.88 |

Geometrical Element | Convection Inside the Building | Convection Outside the Building | Radiation Inside the Studied Room |
---|---|---|---|

Rear wall | yes | - | yes |

Left side wall | yes | - | yes |

Right side wall | yes | - | yes |

Wall under the windows | - | yes | yes |

Columns in the outside wall | - | yes | yes |

Floor | yes | - | yes |

Ceiling | yes | - | yes |

Thermal insulation above the ceiling | - | - | yes |

Wall above the windows | - | yes | yes |

Window frames | - | yes | yes |

Glass in windows | - | yes | yes |

Door leaf | yes | - | yes |

Panel next to the door | yes | - | yes |

Door frame | yes | - | yes |

Door trim | yes | - | yes |

Left heater | - | - | yes |

Right heater | - | - | yes |

**Table 3.**Computed time of the 20 percent temperature decrease. ${t}_{D200}$ is the cooling time for a 200 mm thick wall insulation, ${t}_{D300}$ for a 300 mm thick wall insulation, and ${t}_{D400}$ for a 400 mm thick wall insulation.

${\mathit{c}}_{\mathit{p}}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{J}\xb7{\mathbf{kg}}^{-1}\xb7{\mathbf{K}}^{-1}\right]$ | Room with Windows | Room without Windows | ||||
---|---|---|---|---|---|---|

${\mathit{t}}_{\mathit{D}200}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}300}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}400}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}200}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}300}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}400}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | |

500 | 12.6074 | 13.3499 | 14.2084 | 18.1518 | 22.0157 | 24.5785 |

750 | 12.9352 | 13.7329 | 14.4724 | 19.5924 | 23.7279 | 25.8499 |

1000 | 13.1871 | 13.9521 | 14.6941 | 20.6716 | 24.7484 | 26.4313 |

1250 | 13.3796 | 14.1097 | 14.7164 | 21.4888 | 25.3784 | 26.7341 |

1500 | 13.5251 | 14.3165 | 14.8331 | 22.1175 | 25.7770 | 26.9165 |

1750 | 13.5287 | 14.2861 | 14.9097 | 22.6085 | 26.0335 | 27.0439 |

2000 | 13.7390 | 14.3729 | 14.9442 | 22.9967 | 26.2005 | 27.1444 |

3000 | 13.9666 | 14.4867 | 15.2986 | 23.9202 | 26.4593 | 27.4503 |

4000 | 13.9131 | 14.4992 | 15.3654 | 24.3236 | 26.5035 | 27.6890 |

5000 | 13.9715 | 14.5845 | 15.4119 | 24.5015 | 26.5105 | 27.8799 |

6000 | 14.0064 | 14.5970 | 15.4464 | 24.5761 | 26.5175 | 28.0317 |

**Table 4.**Computed time of the 50 percent temperature decrease. ${t}_{D200}$ is the cooling time for a 200 mm thick wall insulation, ${t}_{D300}$ for a 300 mm thick wall insulation, and ${t}_{D400}$ for a 400 mm thick wall insulation.

${\mathit{c}}_{\mathit{p}}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{J}\xb7{\mathbf{kg}}^{-1}\xb7{\mathbf{K}}^{-1}\right]$ | Room with Windows | Room without Windows | ||||
---|---|---|---|---|---|---|

${\mathit{t}}_{\mathit{D}200}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}300}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}400}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}200}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}300}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | ${\mathit{t}}_{\mathit{D}400}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{h}\right]$ | |

500 | 32.1153 | 33.8433 | 35.9988 | 55.1043 | 80.0684 | 91.1732 |

750 | 32.8130 | 34.6268 | 36.7179 | 63.5432 | 92.1786 | 102.1550 |

1000 | 33.3046 | 35.0553 | 37.2446 | 70.5980 | 99.1117 | 107.5477 |

1250 | 33.6671 | 35.3828 | 37.3249 | 76.1967 | 103.1024 | 110.2212 |

1500 | 33.9439 | 35.7448 | 37.5107 | 80.5065 | 105.3731 | 111.5354 |

1750 | 34.1608 | 35.7814 | 37.6306 | 83.7865 | 106.6239 | 112.1786 |

2000 | 34.3339 | 35.9529 | 37.9464 | 86.2689 | 107.2701 | 112.5012 |

3000 | 34.7661 | 36.7179 | 38.8484 | 91.3301 | 107.5326 | 113.0086 |

4000 | 34.6406 | 36.7179 | 39.0384 | 91.3301 | 107.5326 | 113.0086 |

5000 | 34.7647 | 36.5074 | 39.1732 | 92.6092 | 106.7862 | 114.1834 |

6000 | 34.8449 | 36.5793 | 39.2757 | 92.1753 | 106.7278 | 114.8614 |

**Table 5.**The mesh statistic for determination of desired temperature decrease for a 200 mm (${I}_{200}$) thick wall insulation, for a 300 mm (${I}_{300}$) thick wall insulation, and for a 200 mm (${I}_{400}$) thick wall insulation. The numbers given indicate the number of elements.

Description | Room with Windows | Room without Windows | ||||
---|---|---|---|---|---|---|

${\mathit{I}}_{200}$ | ${\mathit{I}}_{300}$ | ${\mathit{I}}_{400}$ | ${\mathit{I}}_{200}$ | ${\mathit{I}}_{300}$ | ${\mathit{I}}_{400}$ | |

Minimum element quality | 0.01214 | 0.01214 | 0.01214 | 0.09427 | 0.09427 | 0.09427 |

Average element quality | 0.7235 | 0.7239 | 0.7237 | 0.7716 | 0.7738 | 0.7710 |

Tetrahedral elements | 3,062,131 | 3,058,890 | 3,062,773 | 519,500 | 512,976 | 519,485 |

Triangular elements | 378,689 | 378,706 | 378,558 | 45,982 | 46,160 | 46,048 |

Edge elements | 12,220 | 12,203 | 12,184 | 2611 | 2615 | 2611 |

Vertex elements | 276 | 276 | 276 | 112 | 112 | 112 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Charvátová, H.; Procházka, A.; Zálešák, M. Computer Simulation of Temperature Distribution during Cooling of the Thermally Insulated Room. *Energies* **2018**, *11*, 3205.
https://doi.org/10.3390/en11113205

**AMA Style**

Charvátová H, Procházka A, Zálešák M. Computer Simulation of Temperature Distribution during Cooling of the Thermally Insulated Room. *Energies*. 2018; 11(11):3205.
https://doi.org/10.3390/en11113205

**Chicago/Turabian Style**

Charvátová, Hana, Aleš Procházka, and Martin Zálešák. 2018. "Computer Simulation of Temperature Distribution during Cooling of the Thermally Insulated Room" *Energies* 11, no. 11: 3205.
https://doi.org/10.3390/en11113205